Atlas › Modular forms › Weight 2 › Level 731 › 731.2.a › 731.2.a.f

Newform orbit 731.2.a.f

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 2q^{2} - q^{3} + 32q^{4} - 3q^{5} + q^{6} + 5q^{7} + 6q^{8} + 34q^{9} + 12q^{10} + 2q^{11} - 5q^{12} + 26q^{13} - 3q^{14} + 5q^{15} + 62q^{16} - 21q^{17} - 10q^{18} + 16q^{19} - 27q^{20} + 36q^{21} - 14q^{22} - q^{23} + 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} - x^{20} - 86x^{19} + 100x^{18} + 2897x^{17} - 3306x^{16} - 50238x^{15} + 46482x^{14} + 502944x^{13} - 286856x^{12} - 3005123x^{11} + 444674x^{10} + 10310917x^{9} + 2738738x^{8} - 17778836x^{7} - 10981847x^{6} + 11063844x^{5} + 11299712x^{4} + 1516088x^{3} - 1019600x^{2} - 232064x + 8336$

Atkin–Lehner signs

$p$	$w_p$
17	$+1$
43	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	2	−1	−3	5	2	26	−21	16	−1	15	18	12	0	21	−1	39	4	50	16	1	15	56	0	5	−28

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 17, 43, 731]: [0, 2, 4, 68] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 6 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(731, 2)[5]   # 731.2.a.f

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 731:2 1000